Problem: Simplify. Rewrite the expression in the form $z^n$. $\left(z^1\right)^{2}=$
$\begin{aligned} \left(z^1\right)^{2}&=z^{1\cdot 2} \\\\ &=z^{2} \end{aligned}$ This follows from the general rule $\left(x^m\right)^{n}=x^{m\cdot n}$. We can also see this is correct by expanding the powers. $\begin{aligned} \left(z^1\right)^{2}&=\underbrace{z^1\cdot z^1}_\text{2 times} \\\\\\ &=\underbrace{ \underbrace{z}_\text{1 time} \cdot \underbrace{z}_\text{1 time}} _\text{2 times} \\\\ &=z^{2} \end{aligned}$ In conclusion, $\left(z^1\right)^{2}=z^{2}$.